Topology and geometry

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I am attending the four manifolds conference in MN. It seems I really enjoy topology. I do not know if I should consider transfer out of here, or stay to work on invariants with Paul. The choice is not an easy one, because I think I am more algebraic than analytic. But I have intuitive doubt over the power of algebraic methods in topology nowadays. I think a deeper understanding must come from analysis, namely the PDE underlying the geometric structures.

I think I also met some interesting people here. It is very difficult for me to socialize at Binghamton, and relatively easier at here. But I do not really know if I like such experience.

I tried to move on from my thoughts about Leah to other people. I really tried. But I think either I am unlucky or I am just very anti-social. In short I think any more such efforts would be very stupid, as I have to prepare for the qual.

I am really tired. I cannot follow on T.A.Springer, I cannot understand Yau and Schoen, I cannot classify principal bundles, I cannot find a house to live for next year. Sometimes I feel I should quit math and seek life in something else.

I also miss Leah a lot, but what to do? I know I do not love her anymore, and she has never loved me in the first place. The feeling makes me feel frustrated and I do not want to deal with it. I have to read.

It is not clear to me what is the purpose of my graduate school studies. When I was an undergraduate, I chose a huge topic as the goal of my college studies, and my advisor was able to guide me in every crucial turning point because he worked over the same topic 20 years ago when he is an undergraduate in Princeton, and had a better understanding it during his graduate studies in Cambridge and Harvard. I felt fortunate that I had such a great advisor. But I found I am at lost at graduate school, because essentially I do not know what topics I am interested that might coincide with others’ interest, and what might be a good research problem to work on in future.

You have asked me earlier while I was in Bard what my dream is, and I told you the stock response that I wanted to understand the nature of the universe. This is certainly true, only that I did not spend much energy and time to realize it. To me the understanding of nature is something physical; one usually has to propose a theory based on observations and check if the theory matches its predictions. I was rather diffident with my experimental skills and such “checking experimental data” work did really cater to my interest. I was also diffident by the sheer complexity of the physical theory being established. For quite a few subjects, I only know that the laws are true but it is unclear to why they are reasonable in the first place. Further I noticed I could not really getting along in a highly competitive environment where one’s talent relative to others is more important than his or her’s original opinion. I know that nowadays high energy physics is extremely competitive and totally detached from real life. So as a career choice I found studying physics to be incompatible with my personal characteristics. I took a few physics classes at Bard (Modern Physics and Quantum Mechanics), and I found myself feeling quite awkward in these classes. The sheer amount of computation bore me down before revealing me anything interesting, and my lack of patience/diligence did not help either. I do not really know how to tell you about this in the dinning table, because even then it was not entirely clear to me. I still have personal interest in physics, however because my lack of training it is becoming more of a hobby than an integral part of my life.

I found an asylum in mathematics, where I may pursue my interest of understanding the universe abstractly and be detached from people. This is not a progressive choice that one seek deeper understanding in a subject because that fits his/her career goals or personal ambitions. I chose to do mathematics mostly because it is enjoyable, interesting and in my imagination a field required high originality. Also studying mathematics enable me to understand other branches of science rather easily, and I like to associate an abstract theory with something seemingly distant away. For example you know I am interested in mathematical biology, even though I understand little of the subject I found it intellectually appealing that I may understand biological processes via mathematics. The same principle extended over mathematics itself, for which I mean I wish to understand the subject as a whole and not constituted by separated parts.

However, it is not clear to me how to balance the holistic ideals with real life. I am rather busy this semester with all kinds of assignments, and it is becoming increasingly difficult to focus on every field in the same time. Maybe I did not push myself hard enough and wasted too much time. But even if I tried to study very hard, it is clear to me even in fields I have a rather deep understanding, my knowledge is rather superficial due to the level of training I had at Bard. Therefore the only way to solve the problem is to understand deeper rather than expanding wider. But I am in a poor position this semester to study them all deeply enough. While the superficial understanding I achieved at this point is satisfactory to me, they are of little use in practical research and I may forgotten them in a few months without reviewing the material.

The situation is exacerbated because I am running of classes at here, which would not happen in a different university with a larger math department. So while this semester I took 3 classes, next semester I will only have 2 classes available, because 2 of them is in conflict and I can only take one. While I am not really worrying about classes, I am wondering with such a passive choice of learning (taking the highest level classes offering at here) is really optimistic for my future studies. It seems to me the only way to get out of this is to focus on a topic I really wanted to know, and work with an advisor who has more experience in that subject than I do to produce good research. My experience in Bard made me understand the fact that often one does not know what are all the necessary material one need to know until the later stage of research. Then one can naturally expand from his/her research results to other subjects related, and together he/she will have an overall better understanding of the subject.

There are several factors hindering this approach. One is despite of the level of classes I am taking, I am still mathematically immature and need time to decide what my future mathematical interest will be. I emailed Jim Belk (whom you must remember) last semester asking for his advice as I do not have a potential advisor matching my interest at here, and he suggested me to wait one year and expand my interest first. Therefore it seems unrealistic that I should make such decision right now or in next semester.

The other one is more essential. I am interested in the crossroad of three different mathematical subjects, each has a lot of interplay with the other ones. So to do good research on this topic, one need to have a rather deep understanding of almost all related subjects. While a detailed working knowledge of every subject may not be needed, a deep understanding of how them being put up together is almost required. I was fortunate enough to be guided by my advisor and went through some of the associations when I was at Bard. But there are still too much I do not know. This constitute the basic barrier when I want to choose a topic for future studies or consider transferring, because I do not know what I am really interested except some rough idea in the overarching topic. It is possible for me to go to any field of research except really applied ones, for which I have little preparation or knowledge. So I do not really know what to choose for future.

I am planning to make the decision based on my summer school experience. I have applied 3 program and had been accepted by 2 so far (the other one has not send me any decision letter yet). There are in 3 different subjects, and I feel I may have a better understanding of my natural interest if I went through them one by one. But essentially I do not know if this is the right approach. It is clear to me that sometimes one is unhappy with a subject not because the subject is unappealing, but because one formed a rather superficial understanding of it based on limited exposure and little patience. I do not really know if I will like a subject, or a topic if I found some subtopic may be productive for future research.

I am interested in working with Professor P if I chose to stay and need to decide advisor right now. His topic is the very type I mentioned in last paragraph. It is surprisingly deep and very technical. I do not think about these things every day because I am busy with other things, and I could not really talk to anyone about this at here. I really hope I can come up with an interesting topic myself rather than solving someone else’s problems. And it would matter a lot more to me than to get a PhD degree in a higher ranked university but with little creativity output. However, I am unhappy with the low level understanding/training I had in his field of mathematics so far, and it is immature to dream of such things. So I am progressing really slowly this semester, just want to keep a stable pace and not to get lost myself. But maybe I am progressing too slow, for reason I wrote below.

Here is something I could not tell you when you ask me “how are you” every time. I found I cannot focus my attention in work or reading very intensely. This has seldom occured to me and I suspect if I were having obsessive compulsive disorder as my therapist at Bard suggest. For example during the Boston tragedy, which you must have heard and has killed one of Sining’s high school classmates, I could not focus on my studies. I feel anxious as much as everybody else, not that I am particularly worried that my friends may be attacked there, but the suspension of uncertainty made me wondering why this kind of bad thing would happen and how it will end up eventually. And I cannot do work this way as I check CNN periodically. Almost everyday I found a topic “worth” paying attention this way. I found it to be rather damaging my life and my studies, as I often stay up rather late reading meaningless material. I need to find ways to exercise self control and stop this. I do not want to see a therapist again to solve this medically, because I do not want to be addicted. I do not know if this is because of Aspergers, but the syndrome itself does not mean anything and I have to come up with something myself.

I was exhausted this afternoon and went to CIW for dinner. I randomly placed my dish around a person sitting with a dog. Not long afterwards when I finished my meal I found him left with all trash on the table. I went straightforward to warn him to handle his own business, but he refuse to listen. A person jumped out of nowhere urged me to let him go. I was puzzled but granted that nevertheless. He then told me the thug I noticed is in fact a blind student using the dog as a his guidance, and it is to be expected that he cannot find the right place to put the trash. Suddenly I feel very ashamed of myself though I can say I did nothing wrong.

It has been three weeks since school started, and I do not feel I did much. I finished reading some rudimentary commutative algebra notes and read a chapter of Vikal, but this does not really mean anything. I need to quickly catch up with others because what I know soon become obsolete in a few days.

It is not entirely clear to me why understanding a more general concept is better than understanding a simpler, easy concept. I did not thought much about it until today’s index theory class, during which the professor rightly pointed out that index theory in Euclidean spaces are just too simple to delve with. This is certainly true, but when I think about it from a topologist’s point of view, I do not really believe the intrinsic approach working with coordinate patches from a local – global basis is a better route. When we get a much generalized statement that holds for all manifolds, we simultaneously lost the clarity and control we had in Euclidean case. There is certainly the difficulty of writing out an explicit equation of the surface in a Euclidean space make differential geometry be forced to work with the intrinsic as opposed to the extrinsic (Nash Embedding, for example) point of view. But I want to question the merit of using manifold as a concept itself when there are basic questions in Euclidean case needs to be answered.

One way to argue about this kind of generalization is justified is that they are necessary because they are inevitable – no one will disagree that the sphere is a two dimensional surface, and over the projective complex plane the inversion is just a reflection of the inside and outside of the sphere. Another way is to argue that there are simply too many such examples exists in mathematics. What is a line, what is a surface, what is a ball after all? The understanding of dimension is so fundamentally important in all fields of mathematics that a generalization of Euclidean space itself become a powerful theoretical unification language to connect problems which seems totally disconnected from each other. But both points are not really helpful; mathematically there are always a myriad way to generalize a familiar concept, and the fact that one can find easy examples to serve as base cases for an overly generalized concept is usually not supportive of its relative merit in comparison with other concepts. Consider the people living in the flatland world – why would they need to look at the possibility of the existence of the three dimensional space, if they are forever being trapped to the flatland? Why not they focus on their distinct internal problems in flatland first, and then try to move on to other cases?

Let us go back to the situation of a manifold. It seems plain to me that it will not help us to understand the situation of the Euclidean case. Certainly naturally as aforementioned we have seen phase spaces, integral curves, etc all naturally correspond to some kind of topological object that is best to be characterized, even if we are only studying the relative simple case of a linear ODE system like Newtonian mechanics. But there are deep question buried within the Euclidean case; when can two regions in the space be deformed smoothly to each other? What if we impose a smooth condition? What if we impose the maps to be conformal with certain boundary conditions? Can we conclude anything as easily as we defeat the laplace’s equation?

The answer is no, because we do know the situation in dimension 2 (again, thanks to generalizations – Riemann’s mapping theorem), and there are counter examples in dimension 3. But beyond dimension 3 no one knows what is happening. The fact that a relatively simple concept can be properly generalized to a universal object which is harder to analyze but more applicable to real life situations, is the source of the astray; there is a deep difference between the generalization and the original concept in that because of the wiggle room of extra control and extra clearness the problems we left in the base case are extremely hard. It is always nice to look ahead to see a problem that might be be solved via methods totally unexpected in the first place; but beside connecting dots, we also need people who are extremely focused on how to explore further things we thought we know but do not really know very well. This is what I think really important for a mathematican, not because there are already a multitude of people interested in the non-technical side of our lives, but because hard, intricate problems often need deep, innovative ideas to tackle. I still think Euclidean spaces can be a challenge to anyone, in the sense the easy part of the game is finished – and the rest had just begun.

Since I need to do community service, I went to the organization for the orientation program meeting. Unfortunately I came late by 5 minutes because of flat tire. I told the receptionist that I am very sorry, etc, but she told me policy cannot be changed. So I need to go to the meeting in a later date. This make me feel a little dejected, since I hope I can be done with this as soon as possible.

After I went back to school I found life is still very intense. Somanth give me an intense reading assignment from Bott&Tu to present in two weeks, and I have just finished reading commutative algebra. So I have to shift back to topology now. This is fine but I have too much to study at the moment. I managed to finish the commutative algebra notes tonight, and hoping to move on for in depth reading in Eisenbud tomorrow. If I can move by 20 pages a day, in a month I should be done with the majority of the book. But 20 pages a day seems a luxury as Eisenbud contains a lot of problems which I am not sure if I can solve right away.

Somanth also give a list of other reference books to read, and it is too numerous to list at here. It seems I have a lot to learn from now. While this is a good thing I felt deeply unhappy that I did not study as hard as I should when I was at Bard, so my knowledge/skill gap with other people is huge at the moment. There is no short cut and I have to move on very slowly but solidly.

I used to be quite unhappy with the grading process when I was an undergraduate. But now I am being assigned as a grader, with 60+ homework laid in front of me in a huge pile, I realize the task is not that easy. Students makes all kinds of careless mistakes from not noticing the condition given in the problem and omitting totally legitimate solutions. Overall they are not doing very badly but very few did really well. I am wondering if that is the same in other things, where shallow understanding is easy but deeper mastering of the subject remains rare. Mathematics is not the only subject that needs artistic creativity and masonic attention to detail to guarantee full rigor. Otherwise, like one of the erroneous homework solutions I graded – 1^3 could be 2 and we will be living in 1984 instead.